As a dynamic system is considered in this case, in particular, any phenomenon whose time characteristic can be represented in a discrete form of the typex(t+1)=fα(t)(α(t))  (0.1)
Also looked at, however, are systems with several (eg two) simultaneously detected time series x, y according toy(t+τ)=fα(t)(x(t))  (0.2)wherein α (t) is a set of characteristic system parameters, x is a state that generally forms a vector in a multidimensional state space, and y is a state displaced in time. The state space is created by variables that, for example, can be physical, chemical, biological, medical, geological, geometric, numerical and/or process engineering quantities.
The number of system variables that describe the system together with the dynamic response f corresponds to the dimension of the state space. Systems are looked at here whose parameters α may also be variable in time. A given system with parameters α that are invariable in time is also referred to in what follows as a mode.
Observable or measurable system variables (measured quantities) form detectable time series or data streams that are characteristic of the particular sequence of system modes. If the system parameters are invariable for certain time segments within the time series, the time series can be split corresponding to the system modes (segmentation) and each segment can be allocated to a system mode (identification).
Many phenomena in nature as well as in technical applications could be predicted and/or controlled if their basic dynamic processes could be modeled mathematically. The analysis and characterization of practical dynamic systems are often hindered by the fact that the system modes alter while being observed. Examples of this are gradual changes that manifest themselves as drifts or trends of the system parameters, or spontaneous or abrupt changes in the dynamic response of complex systems, for instance when configurations change suddenly, spontaneously or driven from the exterior.
An example of a system considered is the generation of speech signals in the mouth/pharynx region, whereby the system constantly changes its configuration and thus its mode. There is considerable interest in detecting and identifying the modes that are the basis of an observed variable as a function of time (example: fluctuations in air pressure) in order to make better predictions of the system observed or to control it better.
Basically, dynamic systems can be analyzed by measured signals, and a number of methods are known for obtaining models from time series that are suitable for predicting and controlling the response of the system. It is known, for instance, that the state of a dynamic system can be modeled by detecting the time dependence of observed measured quantities. In a first approach this modeling is by reconstruction of the state space by means of so-called time delay coordinates, as described, for example, by N. H. Packard et al. in “Physical Review Letters”, vol. 45, 1980, p 712 ff. Only a single (global) model f for the dynamic response can be found on the basis of such a reconstruction. The global reconstruction of the system is also a disadvantage in that, in applications for multidimensional systems, a large number of input variables must be known in advance as boundary conditions and/or, because of the high dimensionality, the system is virtually impossible to estimate (detect, map) and/or the computing effort is so excessive and quite impractical.
Furthermore, this method is generally inapplicable in the case of parameters that vary with time. The analysis and modeling of dynamic signals are frequently hindered by the fact that the basic systems change with time in essential parameters. Examples are signals in medicine where an organ like the heart or the brain has many dynamic modes that alternate, or speech signals where the generating system, the mouth/pharynx region, apparently adopts different configurations in the course of time.
Another approach is known from the publication by K. Pawelzik, J. Kohlmorgen and K.-R. Mueller in “Neural Computation”, vol. 8, 1996, p 340 ff, where data streams are segmented according to initially unknown system modes changing with time by simulation with several competing models. The models are preferably formed by neural networks, each characteristic of a dynamic response and competing to write the individual points of the data stream by predetermined training rules.
With this method it is possible to break down a time series into segments of quasi-static dynamic response and, simultaneously, to identify models for these system modes from the time series.
Segmentation according to K. Pawelzik et al., details of which are given below, allows allocation of segments to certain system dynamic responses or modes and leads to detection of the data stream as an operation with discrete “switching” between the modes. This description of the parameter dynamic response of complex systems is an advance in terms of accuracy and segmenting different system states compared to the above mentioned global modeling. Nevertheless, the transition between different system states cannot be described satisfactorily. In the analysis of real systems in particular, eg medical applications, it has been found that segmentation is limited to certain cases with mode differences that are as clear as possible and with low noise, and in general is unreliable when the generating system changes with time.
Such changes with time of the generating system make the observable signals transient and mean that the systems, as a rule, can no longer be described by uniform models. If such changes of the system are sudden, one speaks of jump processes.